Reconstructing a non-linear interaction in the dark sector with cosmological observations
RReconstructing a non-linear interaction in the dark sector with cosmologicalobservations
Jiangang Kang ∗ National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, ChinaCAS Key Laboratory of FAST,National Astronomical Observatories,Chinese Academy of Sciences, Beijing 100101, China andSchool of Astronomy and Space Science,University of Chinese Academy of Sciences, Beijing 100049, China
In this work we model two non-linear directly interacting scenarios in dark sector of the universewith the dimensionless parameter α and β , which dominate the energy exchange between darkenergy and dark matter. The central goal of this investigation is to research the interacting modeland discuss the cosmological implications based on the current observational datasets. The classof the interaction is generally characterized by a coupling function Q ∝ H ( z ) ρ x , x denotes theenergy density of dark matter or dark energy. The constrained results we obtained indicate thatthe direct interaction in cosmic dark sector is favored by various observational data and the keyeffects on CMB power spectrum and linear matter power spectrum appear compared to ΛCDMstandard paradigm. Finally, we discuss in depth the effect of different neutrino mass hierarchy onmatter power spectrum and the variation of the ratio of CMB temperature power spectrum C TT(cid:96) and matter power spectrum P ( k ) when the ∆ N eff = N − .
046 from 0.5 to 2, respectively.
Keywords : dark energy; dark matter; cosmological observations; cosmological parameters; neutrinos.
I. INTRODUCTION
The accelerated expansion of the late-time Universe since was discoveried by observations of type Ia supernovae[1, 2], understanding the physical nature behind the acceleration is one of the fundamental topics in moderncosmology.An exotic species of the energy with the equation of state w effd = P d /ρ d < − dubbed as dark energy (DE)has been introduced [3, 4] to explain the expansion of the Universe. Based on Einstein’s theory of GeneralRelativity(GR), the dark energy is represented by the cosmological constant Λ, which combinates with thepressureless cold dark matter (CDM) have regarded as a standard model for modern cosmology.Many predictions from the standard Λ cold dark matter(ΛCDM) theory have been successfully verified bya variety of astronomical observations. However, there are still several challenging puzzles which need to beaddressed, (i) the so-call fine tuning problem, i.e. the result of the vacuum energy density based on quantumtheory reaches some 120 orders of magnitude larger than that inferred from the cosmological constant Λ [5, 6];(ii) coincidence puzzle: is that the energy density of DM and DE are approximate one order of magnitude,although the two quantities do not share similar evolution laws and have different rates of evolution as theuniverse expands, so why do they happen to be of the same order of magnitude right now? [4, 7–12]; (iii) theHubble constant tension: the H = 67 . ± . kms − M pc − derived from Planck 2018( P18) Cosmic MicrowaveBackground (CMB) [13, 14] determination in ΛCDM is less than 4.4 σ error that H = 74 . ± . kms − M pc − obtained directly from the local distance ladder measurements of SN Ia (R19) [15]; (iv) the estimated root-mean-square mass fluctuation amplitude σ in 8 h − M pc spherical volume and matter energy density Ω m from thePlanck 2018 CMB measurements [13, 16] under the ΛCDM cosmology is different from that derived from weaklensing effects based on the KiDS survey [17–19]. Therefore by measuring the growth rate of structure in thedistribution of galaxies as function of redshift, one can place constraints on gravity, and test if dark energycould be due to deviations from GR [20].In order to address these discrepancies, a reasonable solution is recently introduced as a novel mechanismbeyond the canonical ΛCDM model. This modification to dark energy, which is treated as a smooth non-clustering perfect fluid, may directly interact with dark matter without gravity action. Such an idea has beenextensively investigated by a number of phenomenological models, which show that the interaction between DEand DM can alleviate the coincidence problem, and the model predictions have been passed various observationaltests. In particular, many parameterized models assume dynamical couplings between DE and DM, and showthat the current observational measurements support a nonzero interplay behavior for the cosmic dark sector[21– ∗ Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] F e b Q . In thispaper, we propose two novel coupling scenarios of DE and DM, and then put constraints on the relevantparameters within the coupling model Q and Q with the help of mainstream cosmological the combination ofdatasets and finally we research the impact of two interaction models on the C T T(cid:96) and matter power P ( k ) spectraon large scale of universe. Beyond that, we comment on different the neutrino mass hierarchy problem [34–36],i.e. either m < m (cid:28) m , normal hierarchy(NH) or m (cid:28) m < m , inverse hierarchy(IH) and analyse theratios with ΛCDM when setting Σ m ν = 0 . eV and comment on some crucial hints about the neutrino masssplittings can be demonstrated when investigating these IDE models with the current cosmological observations.From the viewpoint of phenomenology, the given assumptions made for Q function should be tested with anumber of cosmological probes [37].This paper is structured as follows: The scenarios of the interaction IDE1 and IDE2 in dark sector we proposeare introduced and from which we derive the relevant evolution equations in Section II. In Section III, we presentthe SN Ia, BAO, CMB and H(z) data combinations adopted in this work as well as the analysis method in eachfitting process with these datasets. The eventual results about the parameters and the C (cid:96) and P ( k ) spectra areobtained from the IDE1 and IDE2 models in Section IV. Finally, we draw our conclusions on two interactingschemes in Section V. We use natural units throughout c = 8 πG = 1. II. THE MODEL OF THE INTERACTION IN DARK SECTOR
Considering a homogeneous and isotropic Universe with a spatially flat Friedmann-Lematre-Robertson-Walker(FLRW) spacetime whose line element follows the form: ds = − dt + a ( t ) (cid:0) dr + r dθ + r sin θdφ (cid:1) , (1)where a ( t ) is cosmic scale factor, t is cosmic time and ( r, θ, φ ) is comoving spherical coordinate system. For auncoupled standard cosmological model (SM) usually contains radiation, baryons, dark matter and dark energy,their perfect fluid energy–momentum tensor satisfies T µν = pg µν + ( ρ + p ) U µ U ν respectively, where U µ,ν standsfor four velocity vector. When introducing the directly interaction of dark constituents, one can understandthe conservation equation of total energy momentum tensor as (cid:53) µ ( T µνc + T µνd ) = 0 based on the Einstein’sfield equation G µν = T cµν + T dµν , c, d denotes dark matter and dark energy while the radiation and baryonsindependently evolve as own mode, respectively. Thus, the FLRW evolution equation for this framework canbe written as 3 H = ρ d + ρ c , (2)2 ˙ H + H = − p d , (3)where an overdot refers to a derivative w.r.t the cosmic time t , H = ˙ a/a is the Hubble parameter, ρ c , ρ d arethe matter density of dark matter and dark energy and p c , p d represents the corresponding pressure of darkmatter and dark energy, respectively. Dark matter is usually treated as a pressureless dust matter, p c = 0 whiledark energy is taken as a negative pressure fluid p d = w d ρ d with the effective equation of state(EoS) w d . Forinstance, if the DE is the smooth vacuum or Λ, i.e. w d = -1. The physical behavior of the effective dark fluidwhen the presence of interaction in dark sector will be different in light of the sign of the coupling function Q .Thus, the whole energy conservation equation for the interaction can be read as the following forms:˙ ρ c + 3 H (1 + w effc ) ρ c = Q (4)˙ ρ d + 3 H (1 + w effd ) ρ d = − Q (5)For a interacting dark fluid of Universe, the sign and strength of the coupling function Q (cid:54) = 0 determines theeffective rate of energy conversion between DM and DE. In the case of Q <
0, meaning that the energy transferfrom dust DM to DE, while for
Q >
0, oppositely which implys the DE likewise decay into pressureless DM.As a phenomenological model, it is usually assumed that the interaction term Q is proportional to the productof energy density of a dark fluid and the Hubble parameter, Q ∝ ρ x H . Conventionally, the coupling strength Q can be tested by a number of observational results with a linear dependence on the energy density of darkelements as presented in the mathematical formula [38–43]: Q = 3 H ( α c ρ c + α d ρ d ) , (6)where α c and α d are dimensionless constants dominating the strength of the DM-DE interaction. In absence ofthe interaction,namely Q = 0, the model restore to ΛCDM cosmology. The redshifted energy density evolutionequations for any coupling term Q and the effective EoS w eff for dark sector can be deduced as: ρ c = ρ c (cid:20)(cid:90) aa ( QaH ) da (cid:21) , (7) ρ d = ρ d (cid:20) a − w d ) − (cid:90) aa ( QaH ) (cid:21) da, (8) w effc = Q Hρ c , w effd = w d − Q Hρ d , (9)where w effc and w effd are respectively termed as the effective equation of state for CDM and DE, ρ c and ρ d denote cold dark matter and dark energy current matter density, and a is today cosmic factor, Q stands forany coupling function of dark sector,respectively,The origin of instabilities depend on different signs of α c and α d and effective EoS w eff under the interactingdark energy (IDE) scenarios[44–49]. In this work, we propose originally two interacting scenarios as followingphysical mechanism: in which the coupling function Q ( t ) can be defined as: IDE Q = 3 Hαρ d (cid:20) βln (1 − ρ c ρ d ) (cid:21) , (10) IDE Q = 3 Hαρ d (cid:20) βsin ( ρ c ρ d ) (cid:21) , (11)where α and β are dimensionless coupling parameter governing the interaction strength of the dark sector andthen one can expands Q function around the ρ c /ρ d = 1 in the manner of Taylor series and the first orderapproximation we take for the interaction model IDE1 and IDE2 and thus Q and Q can be rewritten as anon-linear expression: Q ≈ Hαρ d (1 − βρ c ρ d ) = 3 Hα ( ρ d − βρ c ) , (12)and Q ≈ Hαρ d (1 + βρ c ρ d ) = 3 Hα ( ρ d + βρ c ) , (13)Then we further derive the effective EoS Eq.9 in term of above interaction function via r = ρ c /ρ d for IDE1model: w effc = α ( 1 r − β ) , w effd = w d − α (1 − βr ) , (14)meanwhile for IDE2: w effc = α ( 1 r + β ) , w effd = w d − α (1 + βr ) , (15)Fig.1 shows the evolution behavior of w effc and w effd with respect to redshift z for the two interaction schemes Q and Q ,respectively. The left panel of Fig.1 displays the evolution of w effc as well as the evolution of w effd in the right panel taking same values of the α and β for IDE1 as well as IDE2 when fixing w d = − .
99, onecan find the effective equations of state for dust DM and DE are sensitive to the interaction parameter α and β and the evolution of w effc and w effd for IDE1 and IDE2 maintain stable for z > . w d = − Q < α < β >
0) while
Q > α > β <
0) energy flows from DM to DE after z > .
5, while for IDE2 case, the energy flowsbetween DE and DM opposite direction compared to IDE1 case in term of the various sign of α and β .Now, the main perturbative equations of energy density δ and velocity v under the any interaction Q of DEand DM model in synchronous gauge for dark energy become [38, 50–57]: δ (cid:48) d = a ¯ ρ d Q d − aQ ¯ ρ d (cid:104) δ d + 3 H (cid:16) c s ) d − w d (cid:17) δv (cid:105) , (16) v (cid:48) d = aQ ¯ ρ d (cid:104) v d − (cid:16) c s ) d (cid:17) v d (cid:105) − aρ d (cid:34) c s ) d ρ d a δ d − Qv (cid:35) . (17) z w e ff c IDE
1( = 0.0275, = 0.501)
IDE
1( =0.0275, = 0.501)
IDE
2( = 0.0602, =0.21)
IDE
2( =0.0602, =0.21)
CDM z w e ff d ( w d = . ) IDE
1( =0.018, =0.078)
IDE
1( =0.018, = 0.078)
IDE
2( = 0.019, = 0.053)
IDE
2( = 0.019, = 0.053)
CDM
FIG. 1: The figure shows the redshift evolution of the effective equations of state for DE(top) and CDM (below) underthe interaction models Q and Q when taking different two groups value for α and β ,respectively. Similarly, as for the cold dark matter perturbations, w c = 0 leading to c a ) c = c s ) c = 0, its perturbativeformula of energy δ (cid:48) c and speed v (cid:48) c can be read respectively as: δ (cid:48) c = k v c − h (cid:48) − a ¯ ρ c Q + aQ ¯ ρ c δ c . (18) v (cid:48) c = −H v c − aQ ¯ ρ c ( v − v c ) . (19)here δ i = δρ/ρ ( i = c, d ) and H = aH is comoving Hubble parameter.Besides, we discuss the contribution of matter overdensity δ that resulted from the interaction betweendark energy and dark matter. In term of the general definition of the growth rate of matter perturbations: f c = dlnδ c dlna = δ (cid:48) / δ c and for an arbitrary interaction function Q [30, 58, 59], the growth rate can be understood as: δ (cid:48)(cid:48) c + (cid:18) − QHρ c (cid:19) H δ (cid:48) c = 32 H Ω b δ b + 32 H Ω c δ c (cid:40) ρ d ρ c QHρ c (cid:34) H (cid:48) H , +1 − w d + w d H (1 + w d ) + QHρ c (cid:18) ρ d ρ c (cid:19) (cid:35)(cid:41) , (20)It’s difficult to solve the analysis expression of f c in Eq.20 but we can fit it using the format f ( z ) = Ω γ m ( γ ≈ . Q and coupling factors especiallyafter redshfit z = 2 when DE begins to dominate cosmic energy ingredient and drives the accelerated expansionof the universe .In this work, we investigate cosmological dynamics with the flat spacetime due to the above Eqs.1–13, ne-glecting the radiation density Ω r ≈ − , therefore assuming the matter density relation Ω m +Ω d = 1 for IDE1case can be rewritten: H ( z ) /H = Ω d (cid:18) wα + β (1 + z ) + 1 α + β (1 + z ) w + α ) (cid:19) + Ω m (1 + z ) , (21)For the IDE2 scheme: H ( z ) /H = Ω m (cid:18) wα + β (1 + z ) w ) + 1 α + β (1 + z ) − w − α ) (cid:19) + Ω d (1 + z ) w ) (22)where Ω m and Ω d are current energy density of dark matter and dark energy. z f IDE
1( = 0.406, = 0.074)
CDM ( = 0.55)
IDE
2( = 0.501, = 0.12)
FIG. 2: The linear evolution of fσ as the function of redshift z for interaction models Q and Q (dash blue and limeline,black solid line stands for ΛCDM model, respectively). Errorbar datapoints are employed in Table I of [62] III. DATA AND METHODOLOGY
In this section we introduce the mainstreams cosmological probes and the likelihood function
L ∝ e − χ / used in this work to extract the posterior distribution functions (PDFs) of each parameter within IDE1 andIDE2 framework. A. SN Ia data
The Type Ia Supernovae (SN Ia) play a key role in unveiling the character of the dark energy as the ”standardcandles”. We utilize the Pantheon compilation consisting of 1048 SN Ia samples covering redshift range 0 . Table.I lists the adopted quantities that derived from the different BAO measurements as the ”standardruler”, r s is the radius of the comoving sound horizon at the drag epoch as following: r s = (cid:90) t s c s d ta , (26) TABLE I: The 12 BAO data used in this work redshift from z = 0 . r s , fid Survey Reference0.106 r s /D V ± r s /D V ± r s /D V ± r s /D V ± r s /D V ± r s /D V ± r s /D V ± r s /D V ± r s /D V ± D M ( r s , fid /r s ) 1518 ± 22 147.78 SDSS DR12 [71]0.51 D M ( r s , fid /r s ) 1977 ± 27 147.78 SDSS DR12 [71]2.40 D H /r s ± where c s is the sound speed, t s is the epoch of last scattering, a is the scale factor. Since r s are not sensitive tophysics at low redshifts, we fix r s = 147 . ± . 26 measured from the Planck 2018 release [13]. The sphericallyaveraged distance D V is given by D V ( z ) = (cid:20) D ( z ) czH ( z ) (cid:21) / , (27)where D M ( z ) = (1 + z ) D A is the comoving angular diameter distance, and D A = D C / (1 + z ) is the physicalangular diameter distance. D H = c/H ( z ) is the Hubble distance.The χ for the BAO data can be calculated by χ = ∆ D T · C D − · ∆ D , (28)where ∆ D = D obs − D th , D obs and D th are the observational and theoretical quantities shown in Table . I and C D is corresponding covariance matrix. C. CMB data For the CMB dataset, we exploit the parameter of acoustic scale l A , the shift parameter R , the decouplingredshift z ∗ inferred from the Planck CMB measurement and the distance priors from the Planck 2018 results[13,73], they are defined respectively as R = (cid:113) Ω m H r ( z ∗ ) , (29) (cid:96) A = πr ( z ∗ ) /r s ( z ∗ ) , (30)where Ω m is current energy density of dark matter. r ( z ∗ ) is the comoving size of the sound horizon at theredshift of the decoupling epoch of photons z ∗ , z ∗ can be expressed as [74–76]: z ∗ = 1048[1 + 0 . b h ) − . ][1 + g (Ω m h ) g ] , (31)where g = 0 . b h ) − . . b h ) . , g = 0 . . b h ) . . (32)The χ distribution for the CMB data can be estimated as χ = ∆ X T C − ∆ X , (33)where ω b = Ω b h is the baryon energy density, h is the dimensionless Hubble constant. ∆ X = X obs − X th , X = ( R , (cid:96) A , ω b ) and C CMB is the covariance matrix. TABLE II: The table lists the priors on the parameters space.Parameter Prior rangeΩ m [0, 0.7] h [0.5, 1] α [-1, 1] β [-1, 1] w effd [-3,-0.3] σ [0.2,1.4]TABLE III: The table summarizes the mean values and 1- σ (68.3%) uncertainties of the parameters under the scenarioof the IDE1: Q = 3 Hαρ d (1 + βln (1 − ρ c ρ d ) case in Fig.3. Here the parameter Ω m equal the energy fractions of baryonsplus dark matter, i.e. Ω m = Ω c + Ω b .Parameter CMB CMB+SN Ia CMB+BAO+H(z) CMB+SN Ia+BAO+H(z)Ω m . +0 . − . . +0 . − . . +0 . − . . +0 . − . h . +0 . − . . +0 . − . . +0 . − . . +0 . − . α − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . β . +0 . − . . +0 . − . . +0 . − . . +0 . − . w effd − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ red D. H(z) Data The H ( z ) data contain 30 data points in the redshift range from 0 to 2 in Table 4 of [77], which are ob-tained using the differential-age method, that is to compare the ages of passively-evolving galaxies with similarmetallicity, separated in a small redshift interval [78] serving as cosmic chronometers and yielding a model-independent measurement of the expansion rate of the Universe at various redshifts. The χ distribution forthe H ( z ) data can be expressed as [75, 76] χ H = N =30 (cid:88) i =1 [ H obs ( z i ) − H th ( z i )] σ H . (34)Here H obs and H th are the observational and theoretical Hubble parameters, respectively, and σ H is the error.Finally, the joint χ of above four datasets employed in this work can be calculated by χ = χ SN + χ BAO + χ CMB + χ H ( z ) . (35) IV. RESULTS In order to analyse the IDE1 and IDE2 model introduced in Sec.II and place constraints on the free parameterswithin the scenarios, we use CMB data and the combination of CMB+Pantheon, CMB+BAO+H(z), andCMB+Pantheon++BAO+H(z) datasets and modify the Boltzmann code CLASS [79] to solve the backgroundand perturbation equations of IDE1 and IDE2 model and then we utilize MontePython, a Markov chain MonteCarlo package (MCMC)[80–82] with the Metropolis-Hastings algorithm in accordance with the Gelman-Rubinconvergence criterion, requiring | R − | < χ red = χ min / ( N − n ) in the fitting process for each dataset combination, where χ min is the minimum https://github.com/lesgourg/class_public https://github.com/brinckmann/montepython_public h -0.483-0.472-0.461-0.451 α β -1.09-1.06-1.03-0.996 w σ Ω m σ h -0.483 -0.472 -0.461 -0.451 α β -1.09 -1.06 -1.03 -0.996 w CMBCMB+SN IaCMB+BAO+H(z)CMB+SN Ia+BAO+H(z) FIG. 3: The one-dimensional posterior distributions and two-dimensional contours in 1- σ (68.3%) and 2- σ (95.5%)confidence level(C.L.) of the free parameters for the Q = 3 Hαρ d (1 + βln (1 − ρ c ρ d ) interacting dark energy scenario. Thered, blue, green, and orange plots stand for the constraint results from CMB, CMB+SN Ia, CMB+BAO+H(z) and jointdatasets, respectively. χ , N and n are the number of data point and free parameter,respectively. The priors on the parameter spaceare shown in Table.II.Firstly, Fig.3 and Fig.4 show the 68.3% and 95.5% confidence level(C.L.) contour map and 1-D PDFs ofthe free parameter of IDE1: Q = 3 Hαρ d (1 + βln (1 − ρ c ρ d ) and IDE2: Q = 3 Hαρ d (1 + βsin ( ρ c ρ d )) modelsexplored by using CMB and CMB+SN Ia and CMB+BAO+H(z) and joint datasets respectively because ofthese datasets can break the degeneracy of the among quantities. We find that the model can be constrainedby these datasets well due to χ red is close to 1 for the above four data combinations. As listed on table III,the best-fit value of the coupling factor α and β in Fig 3 yield α (cid:39) − . +0 . − . and β (cid:39) . +0 . − . for thejoint datasets and the Q ≈ − . H ( z ) < w effd < − Q = 3 Hαρ d (1 + βsin ( ρ c ρ d )) scenario, the mean values of the interaction factors α (cid:39) − . β (cid:39) . 775 and so Q ≈ . H ( z ) > H tension or the σ discrepancy [16, 83] mentioned in Section.I andboth issues can be slightly alleviate due to the constraints values under the two interacting IDE1 and IDE2framework. From Table.III we see, the IDE1 model gives the dimensionless Hubble constant h from 0.666 to h -0.88-0.87-0.86-0.85 α β -1.09-1.06-1.03-0.996 w σ Ω m σ h -0.88 -0.87 -0.86 -0.85 α β -1.09 -1.06 -1.03 -0.996 w CMBCMB+SN IaCMB+BAO+H(z)CMB+SN Ia+BAO+H(z) FIG. 4: The one-dimensional posterior distributions and two-dimensional contours in 1- σ (68.3%) and 2- σ (95.5%)confidence level(C.L.) of the free parameters for the Q = 3 Hαρ d (1 + βsin ( ρ c ρ d )) ρ c interacting dark energy scenario. Thered, blue, green, and orange plots show the constraint results from CMB, CMB+SN Ia, CMB+BAO+H(z) and jointdatasets, respectively. σ level and similarly, the resultingvalues σ are tightly obtained from 0.811 to 0.827. While in the IDE2 model, the tensions are also decreasedto some extent, the values of h and σ show no serious deviation from the CMB data only to the joint datasetfitting. In short, these results of parameters from solely Planck CMB data to four observations combinationsimply that the non-linear interactions of DE and DM considered in this work can be supported well by theavailable observational data which determined from ΛCDM cosmology.In order to further understand the relation between the coupling factor α and β and the time of matter-radiation equality for the existing of an interaction in cosmic dark sector, we draw the evolution curves ofΩ m / Ω r in redshift space z , the time of matter-radiation equality comes later in both IDE1 and IDE2 case withincreasing value of α and β around redshift z =3400 to 3500 compared to non-interacting model as shown inFig.5, indicating the increasing of interaction parameter α and β will lead to a addition of energy density of thedust matter Ω m , the epoch of matter-radiation balance will become late and the sound horizon will scales up.Fig.6 presents some key signatures on the CMB temperature power spectrum in left and the linear matterpower spectrum in right panel when taking the different values for α and β describing interacting models IDE1and IDE2. Concerning the CMB temperature power spectra in the left panel, we deem that actually either darkenergy or dark matter gains additional energy density and influences the microwave background temperatureanisotropy especially at low multiple l < 30 under the IDE1 and IDE2 respectively. The increase of dark energydensity within IDE1 compared to IDE2, especially alters integrated Sachs-Wolfe(ISW) level on large scale dueto the decay of gravitational potential over expansion history as well as changes era of matter-radiation balance0 TABLE IV: The table shows mean values and 1- σ (68.3/%) confidence level(C.L.) of the cosmological parameters underthe IDE2: Q = 3 Hαρ d (1 + βsin ( ρ c ρ d )) case in Fig.4 fitted by above datasets,here Ω m = Ω c + Ω b .Parameter CMB CMB+SN Ia CMB+BAO+H(z) CMB+SN Ia+BAO+H(z)Ω m . +0 . − . . +0 . − . . +0 . − . . +0 . − . h . +0 . − . . +0 . − . . +0 . − . . +0 . − . α − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . β . +0 . − . . +0 . − . . +0 . − . . +0 . − . w effd − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ min z m / r ( w d = . ) IDE1( = 0.026, = 0.017)IDE1( = 0.163, = 0.316)IDE2( = 0.409, = 0.014)IDE2( = 0.553, = 0.308) FIG. 5: The evolution for the ratio of matter and radiation Ω m and Ω r with the variation of the interaction parameter α and β for IDE1 and IDE2, here Ω m = Ω c + Ω b . The horizontal gray thick line corresponds to the case of Ω m = Ω r ,namely the era of the matter-radiation equality and we fix these parameters for the plot Ω m = 0 . b = 0 . d =0 . r = 0 . ( + ) / C TT [ u k ] CDMIDE1( =0.024, =0.073)IDE2( = 0.018, = 0.009) 10 k [ h /Mpc]10 P ( k )[ M p c / h ] CDMIDE1( =0.017, = 0.051)IDE2( = 0.043, =0.045) FIG. 6: The effects on the CMB temperature power spectra(left panel) and the matter power spectra (right panel)fortwo case of the interaction term Q and Q . The red and blue curves lines for the IDE1: Q and IDE2: Q reference toΛCDM in green solid line, respectively. Multipole C TT / C TT , C D M ( I D E ) CDM ( N eff = 3.046) N eff = 0.5 N eff = 1 N eff = 1.5 N eff = 2 Multipole C TT / C TT , C D M ( I D E ) CDM ( N eff = 3.046) N eff = 0.5 N eff = 1 N eff = 1.5 N eff = 2 FIG. 7: The ratio of the CMB temperature power spectra to ΛCDM in presence of neutrino for the interaction termIDE1(left panel) and IDE2(right panel) with the variation of ∆ N eff from 0.5 to 2. k [ h /Mpc] P ( k ) / P ( k ) C D M ( I D E ) CDM ( N eff = 3.046) N eff = 0.5 N eff = 1 N eff = 1.5 N eff = 2 k [ h /Mpc] P ( k ) / P ( k ) C D M ( I D E ) CDM ( N eff = 3.046) N eff = 0.5 N eff = 1 N eff = 1.5 N eff = 2 FIG. 8: The ratio of the matter power spectra to ΛCDM of the interaction term IDE1(left panel) and IDE2(right panel)when ∆ N eff from 0.5 to 2. and the height of the first peak of CMB temperature power spectra is lifted for the tight coupling of darksector. As for the right panel, the dominant effects on the linear matter power spectrum P ( k ) show up atlarge scales( k < . h/M pc ), hinting the enhancement of the content of dark matter that forms larger cosmicstructures and earlier matter clusters of galaxies under its gravity. Consequently, the shape of matter powerspectra P ( k ) somewhat is more intensified due to larger horizon boundary in IDE2 than IDE1 model. Moreover,the first peak evidently is rised for IDE1 in small multipole (cid:96) ¡10 since the dark energy fluid flows to dark matterto change the energy of CMB photons when across it. The presence of interaction of dark sector changes theregular evolution law and which is imprinted on the integrated Sachs-Wolfe (ISW) effect in large scale effect ofthe CMB TT spectrum.In this paragraph, we comment on neutrino hierarchies problem under the framework IDE1 and IDE2 model.The experiment of neutrino oscillation has suggested that neutrinos have mass size and mass splittings amongthree flavors neutrino, though either absolute mass of neutrinos or mass hierarchies remian puzzling(see [31,34, 35, 37, 84] for a review). Actually, both the properties of massive neutrinos and dark energy could imprintimportant signatures on expansion history and large scale structure (LSS) in the evolution history of theuniverse[35, 36, 85, 86, 86]. Therefore, It’s expected that some significant information can be found on C (cid:96) and P ( k ) spectrum in the direct interaction of dark energy and dark matter scenario when considering masshierarchies. From the ratio of C T T(cid:96) to C T T(cid:96) in ΛCDM model in Fig.7, one can find there is mass size and masshierarchies of neutrino species and showing bigger deviation in IDE1 than IDE2 related to ΛCDM regardless ofNH or IH mode and demonstrating the gradual suppression for the larger free degree of ∆ N eff = N − . 046 from0.5 to 2 occurring in IDE1 and IDE2 due to the smoothing effect of more neutrinos free screaming. Similarly,concerning the ratio of matter power P ( k ) /P ( k ) Λ CDM is dramatically intensified in k > . h/M pc in the2wake of ∆ N eff turns bigger from 0.5 to 2 in Fig.8, the attribute of neutrinos may strongly degeneracy withdark matter halos to cluster of baryon particles and host a vast of galaxies or other objects in both IDE1 andIDE2 models, but in k < . h/M pc phase, the magnitude of ratio amplitude of IDE2 than IDE1 is less thanand tend to 1 due to the density of neutrinos are diluted since in larger scales and difficult to cluster. V. CONCLUSIONS In this analysis, we have investigated two scenarios of direct interaction between the dark energy and darkmatter, in which the interaction term Q is reconstructed as Eq.10 and Eq.11 with coupling parameter α and β determining the conversion between DE and DM.Firstly, we come up with two novel interacting model IDE1 and IDE2, and then we use (i) the cosmicmicrowave background from Planck 2018 result, (ii) Pantheon sample of Supernovae Type Ia, (iii) baryonacoustic oscillations distance measurements and (iv) direct H(z) observation to put constraints on the twointeracting scenarios Q = 3 Hαρ d (1 + βln (1 − ρ c ρ d ) and Q = 3 Hαρ d (1 + βsin ( ρ c ρ d )) by calling the MontePythonsampler and Class code. All results summarized in Table.III and IV and Fig.3 and 4, we find that(1) our analysis with all combined datasets support non-zero value for the interacting factors α and β whilethe dark energy equation of state(EoS) w < − σ confidence level.(2) With IDE1 or IDE2 model, we point out that the results from the interactions are compatible with a generalmodel such as the uncoupled w CDM cosmology and fit the latest observations. Specially, it is evident that theinteracting models relaxed the existing tensions of H and σ arised from different cosmological measurementsrelying on the ΛCDM picture at about 2 σ confidence level, though it does not seem to be able to completelyalleviate them.(3) When the addition of neutrinos mass normal hierarchy(NH) and inverse hierarchy(IH) mode to theinteraction theory of IDE1 and IDE2 scenario, we discuss that the influence on the ratio of CMB temperaturespectrum, indicating the neutrinos obviously smooth temperature fluctuation especially in k > 100 with thefree degree error ∆ N eff from 0.5 to 2, while the effect of dark matter to drag baryons particles to cluster andspark the physical processes are reinforced due to the more neutrinos behaving like dark matter to form largerstructures of halos.As a whole, we confirm that the interaction of dark energy and dark matter in this work can be explored bythe cosmological observation combinations and eliminate some discrepancies to some extent that from theoryand determination and even can produce some key signals on the C (cid:96) and P ( k ) spectrum can be detected in anumber of cosmology surveys in future. Acknowlegment We thank Tongjie Zhang for constructive comments on the manuscript. [1] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan,S. Jha, R. P. Kirshner, et al., Astron. J. , 1009 (1998), astro-ph/9805201.[2] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar,D. E. Groom, et al., Astrophys. J. , 565 (1999), astro-ph/9812133.[3] P. J. Peebles and B. Ratra, Reviews of Modern Physics , 559 (2003), astro-ph/0207347.[4] E. J. Copeland, M. Sami, and S. Tsujikawa, International Journal of Modern Physics D , 1753 (2006), hep-th/0603057.[5] M. Li, X.-D. Li, S. Wang, and Y. Wang, Communications in Theoretical Physics , 525 (2011), 1103.5870.[6] C. Wetterich, Astron. Astrophys. , 321 (1995), hep-th/9408025.[7] B. Wang, Y. Gong, and E. Abdalla, Physics Letters B , 141 (2005), hep-th/0506069.[8] D. Tocchini-Valentini and L. Amendola, Phys. Rev. D , 063508 (2002), astro-ph/0108143.[9] L. P. Chimento, A. S. Jakubi, D. Pav´on, and W. Zimdahl, Phys. Rev. D , 083513 (2003), astro-ph/0303145.[10] P. Rudra, in (2016), vol. 41 of COSPAR Meeting , pp. H0.2–11–16.[11] H. Paul and M. Pavicic, Journal of the Optical Society of America B Optical Physics , 1275 (1997), quant-ph/9908023.[12] I. Zlatev, L. Wang, and P. J. Steinhardt, Phys. Rev. Lett. , 896 (1999), astro-ph/9807002.[13] P. Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday,R. B. Barreiro, N. Bartolo, et al., Planck 2018 results. vi. cosmological parameters (2018), 1807.06209. [14] E. Di Valentino, Nature Astronomy , 569 (2017), 1709.04046.[15] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J. , 85 (2019), 1903.07603.[16] M. Kilbinger, L. Fu, C. Heymans, F. Simpson, J. Benjamin, T. Erben, J. Harnois-D´eraps, H. Hoekstra, H. Hildebrandt, T. D. Kitching, et al., Mon. Not. R. Astron. Soc. , 2200 (2013), 1212.3338.[17] F. K¨ohlinger, M. Viola, B. Joachimi, H. Hoekstra, E. van Uitert, H. Hildebrandt, A. Choi, T. Erben, C. Heymans,S. Joudaki, et al., Mon. Not. R. Astron. Soc. , 4412 (2017), 1706.02892.[18] H. Hildebrandt, M. Viola, C. Heymans, S. Joudaki, K. Kuijken, C. Blake, T. Erben, B. Joachimi, D. Klaes, L. Miller,et al., Mon. Not. R. Astron. Soc. , 1454 (2017), 1606.05338.[19] S. Joudaki, C. Blake, A. Johnson, A. Amon, M. Asgari, A. Choi, T. Erben, K. Glazebrook, J. Harnois-D´eraps,C. Heymans, et al., Mon. Not. R. Astron. Soc. , 4894 (2018), 1707.06627.[20] L. Guzzo, M. Pierleoni, B. Meneux, E. Branchini, O. Le F`evre, C. Marinoni, B. Garilli, J. Blaizot, G. De Lucia,A. Pollo, et al., Nature (London) , 541 (2008), 0802.1944.[21] V. Salvatelli, N. Said, M. Bruni, A. Melchiorri, and D. Wands, Physical Review Letters , 181301 (2014),1406.7297.[22] W. Yang and L. Xu, Phys. Rev. D , 083532 (2014), 1409.5533.[23] W. Yang and L. Xu, Phys. Rev. D , 083517 (2014), 1401.1286.[24] R. C. Nunes, S. Pan, and E. N. Saridakis, Phys. Rev. D , 023508 (2016), 1605.01712.[25] C. van de Bruck, J. Mifsud, and J. Morrice, Phys. Rev. D , 043513 (2017), 1609.09855.[26] W. Yang, H. Li, Y. Wu, and J. Lu, JCAP. , 007 (2016), 1608.07039.[27] C. van de Bruck and C. C. Thomas, Phys. Rev. D , 023515 (2019), 1904.07082.[28] W. Yang, S. Pan, and A. Paliathanasis, Mon. Not. R. Astron. Soc. , 1007 (2019), 1804.08558.[29] M. Martinelli, N. B. Hogg, S. Peirone, M. Bruni, and D. Wands, Mon. Not. R. Astron. Soc. , 3423 (2019),1902.10694.[30] S. Pan, W. Yang, and A. Paliathanasis, Monthly Notices of the Royal Astronomical Society , 3114 (2020),ISSN 0035-8711, https://academic.oup.com/mnras/article-pdf/493/3/3114/32890756/staa213.pdf, URL https://doi.org/10.1093/mnras/staa213 .[31] X.-J. Bi, B. Feng, H. Li, and X. Zhang, Phys. Rev. D , 123523 (2005), hep-ph/0412002.[32] R.-Y. Guo, J.-F. Zhang, and X. Zhang, Chinese Physics C , 095103 (2018), 1803.06910.[33] C. van de Bruck and J. Mifsud, Physical Review D (2018), ISSN 2470-0029, URL http://dx.doi.org/10.1103/PhysRevD.97.023506 .[34] S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goobar, and O. Mena, Phys. Rev. D , 083501 (2018),1801.08553.[35] S. Betts, W. R. Blanchard, R. H. Carnevale, C. Chang, C. Chen, S. Chidzik, L. Ciebiera, P. Cloessner, A. Cocco,A. Cohen, et al., arXiv e-prints arXiv:1307.4738 (2013), 1307.4738.[36] J. Lesgourgues and S. Pastor, 307 (2006), astro-ph/0603494.[37] L. Feng, H.-L. Li, J.-F. Zhang, and X. Zhang, Science China Physics, Mechanics, and Astronomy , 220401 (2020),1903.08848.[38] M. B. Gavela, D. Hern´andez, L. Lopez Honorez, O. Mena, and S. Rigolin, JCAP. , 034 (2009), 0901.1611.[39] M. Quartin, M. O. Calv˜ao, S. E. Jor´as, R. R. R. Reis, and I. Waga, JCAP. , 007 (2008), 0802.0546.[40] C. G. B¨ohmer, G. Caldera-Cabral, R. Lazkoz, and R. Maartens, Phys. Rev. D , 023505 (2008), 0801.1565.[41] C. B¨ohmer, G. Caldera-Cabral, R. Lazkoz, and R. Maartens, in American Institute of Physics Conference Series ,edited by K. E. Kunze, M. Mars, and M. A. V´azquez-Mozo (2009), vol. 1122 of American Institute of PhysicsConference Series , pp. 197–200.[42] W. Zimdahl and D. Pav´on, Classical and Quantum Gravity , 5461 (2007), astro-ph/0606555.[43] R. R. Bachega, A. A. Costa, E. Abdalla, and K. Fornazier, Journal of Cosmology and Astroparticle Physics ,021–021 (2020), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2020/05/021 .[44] Y.-H. Li, J.-F. Zhang, and X. Zhang, Phys. Rev. D , 063005 (2014), 1404.5220.[45] Y.-H. Li, J.-F. Zhang, and X. Zhang, Phys. Rev. D , 123007 (2014), 1409.7205.[46] R.-Y. Guo, Y.-H. Li, J.-F. Zhang, and X. Zhang, JCAP. , 040 (2017), 1702.04189.[47] X. Zhang, Science China Physics, Mechanics, and Astronomy , 50431 (2017), 1702.04564.[48] J.-P. Dai and J.-Q. Xia, Astrophys. J. , 125 (2019), 1904.04149.[49] W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi, and D. F. Mota, JCAP. , 019 (2018), 1805.08252.[50] C.-P. Ma and E. Bertschinger, Astrophys. J. , 7 (1995), astro-ph/9506072.[51] B. M. Jackson, A. Taylor, and A. Berera, Phys. Rev. D , 043526 (2009), 0901.3272.[52] M. B. Gavela, L. Lopez Honorez, O. Mena, and S. Rigolin, JCAP. , 044 (2010), 1005.0295.[53] V. Salvatelli, A. Marchini, L. Lopez-Honorez, and O. Mena, Phys. Rev. D , 023531 (2013), 1304.7119.[54] L. Feng, J.-F. Zhang, and X. Zhang, Physics of the Dark Universe , 100261 (2019), 1712.03148.[55] G. Ballesteros and J. Lesgourgues, JCAP. , 014 (2010), 1004.5509.[56] J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao, and X. Zhang, International Journal of Modern Physics D , 1229(2008), astro-ph/0703202.[57] R. J. F. Marcondes, R. C. G. Landim, A. A. Costa, B. Wang, and E. Abdalla, JCAP. , 009 (2016), 1605.05264.[58] J. B. Jim´enez, L. Heisenberg, T. Koivisto, and S. Pekar, Phys. Rev. D , 103507 (2020), 1906.10027.[59] B. J. Barros, T. Barreiro, T. Koivisto, and N. J. Nunes, Physics of the Dark Universe , 100616 (2020), 2004.07867.[60] E. V. Linder and R. N. Cahn, Astroparticle Physics , 481 (2007), astro-ph/0701317.[61] M. J. Hudson and S. J. Turnbull, The Astrophysical Journal , L30 (2012), ISSN 2041-8213, URL http://dx. doi.org/10.1088/2041-8205/751/2/L30 .[62] B. Sagredo, S. Nesseris, and D. Sapone, Phys. Rev. D , 083543 (2018), 1806.10822.[63] D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan, R. Chornock, R. J. Foley, M. E. Huber, R. Kessler, G. Narayan,A. G. Riess, et al., Astrophys. J. , 101 (2018), 1710.00845.[64] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, andF. Watson, Mon. Not. R. Astron. Soc. , 3017 (2011), 1106.3366.[65] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden, and M. Manera, Mon. Not. R. Astron. Soc. ,835 (2015), 1409.3242.[66] L. Anderson, ´E. Aubourg, S. Bailey, F. Beutler, V. Bhardwaj, M. Blanton, A. S. Bolton, J. Brinkmann, J. R.Brownstein, A. Burden, et al., Mon. Not. R. Astron. Soc. , 24 (2014), 1312.4877.[67] N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta, and E. Kazin, Mon. Not. R.Astron. Soc. , 2132 (2012), 1202.0090.[68] C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch, S. Croom, D. Croton, T. M. Davis, M. J. Drinkwater,K. Forster, et al., Mon. Not. R. Astron. Soc. , 405 (2012), 1204.3674.[69] T. Delubac, J. E. Bautista, N. G. Busca, J. Rich, D. Kirkby, S. Bailey, A. Font-Ribera, A. Slosar, K.-G. Lee, M. M.Pieri, et al., Astron. Astrophys. , A59 (2015), 1404.1801.[70] S. W. Hell, S. J. Sahl, M. Bates, X. Zhuang, R. Heintzmann, M. J. Booth, J. Bewersdorf, G. Shtengel, H. Hess,P. Tinnefeld, et al., Journal of Physics D Applied Physics , 443001 (2015), 1711.04999.[71] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-H.Chuang, et al., Mon. Not. R. Astron. Soc. , 2617 (2017), 1607.03155.[72] H. du Mas des Bourboux, J.-M. Le Goff, M. Blomqvist, N. G. Busca, J. Guy, J. Rich, C. Y`eche, J. E. Bautista,´E. Burtin, and K. S. Dawson, Astron. Astrophys. , A130 (2017), 1708.02225.[73] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J.Banday, R. B. Barreiro, N. Bartolo, et al., Astron. Astrophys. , A5 (2020), 1907.12875.[74] W. Hu and N. Sugiyama, Astrophys. J. , 542 (1996), astro-ph/9510117.[75] S. Cao, N. Liang, and Z.-H. Zhu, Monthly Notices of the Royal Astronomical Society , 1099–1104 (2011), ISSN0035-8711, URL http://dx.doi.org/10.1111/j.1365-2966.2011.19105.x .[76] R. Lazkoz and E. Majerotto, Journal of Cosmology and Astroparticle Physics , 015–015 (2007), ISSN 1475-7516,URL http://dx.doi.org/10.1088/1475-7516/2007/07/015 .[77] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro, andD. Wilkinson, Journal of Cosmology and Astroparticle Physics , 014–014 (2016), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2016/05/014 .[78] R. Jimenez and A. Loeb, Astrophys. J. , 37 (2002), astro-ph/0106145.[79] D. Blas, J. Lesgourgues, and T. Tram, JCAP. , 034 (2011), 1104.2933.[80] B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, JCAP , 001 (2013), 1210.7183.[81] T. Brinckmann and J. Lesgourgues (2018), 1804.07261.[82] H. G. Katzgraber, Introduction to monte carlo methods (2011), 0905.1629.[83] E. Di Valentino and S. Bridle, Symmetry (2018), ISSN 2073-8994, URL .[84] S. F. King, Contemporary Physics , 195 (2007), https://doi.org/10.1080/00107510701770539, URL https://doi.org/10.1080/00107510701770539https://doi.org/10.1080/00107510701770539